the ultimate importance of invariant property : rothalpy
TRANSCRIPT
The Ultimate Importance of Invariant Property : Rothalpy
Another Crucial Zone of Concern
Flow in Stator-Rotor Inter stage Gaps
Geometrical Details along the Third Direction
• True flow through a turbo-machinery is three-dimensional.
• Flow and tangential flow velocities are very important for better operation of a turbo-machine.
• The third component, which is normal to flow and tangential directions is in general of no use.
• This direction can better represented as blade height direction.
Third Direction of an Axial Flow Turbo-Machines
• The third direction in an axial flow machine is the radial direction.
• The direction of Centrifugal forces!
• Strong centrifugal forces are exerted on blades & fluid in radial direction.
• The centrifugal field distorts the flow velocity profiles considerably.
• Fluid particles tend to move outwards rather than passing along cylindrical stream surfaces as classically assumed.
• Particularly in tall blade (low hub: tip) ratio designs.
• An approach known as the radial equilibrium method, widely used for three-dimensional design calculations in a an axial flow machine.
Radial Equilibrium Theory of Turbo-machines
P M V SubbaraoProfessor
Mechanical Engineering Department
A Model for Stable Operation of A Machine
A guiding equation for distribution of load along blade length ….
Radial Variation of Blade Geometry
Radial Equilibrium Theory
• Assumes that flow is in radial equilibrium before and after a blade row.
• Radial adjustment takes place through the row.
• More important for Axial Flow Machines.
Radial Equilibrium Analysis
The centrifugal force = (rdrd)2r V = r
The centrifugal force is
The pressure force on the element
drdVF lcentrifuga2
rdpdFpressure
If the two forces are the only ones acting (viscous and other effects neglected), the particle will move at constant radius if:
lcentrifugapressure FF
r
drVdp 2
drdVrdpd 2
Equilibrium Condition for A Rotating Fluid
An equivalent equation for compressible flow can be developed by using the following thermodynamic relation:
0dp
dhvdpdhTds
dp
dh r
drVdh 2
The radial variation of whirl velocity should be according to above equation.
How to implement on a machine?
2222
2222
0VVV
hV
hh rf
0222
222
0
VVV
hddh rf
Total Energy Equation for A Rotating Fluid
Stagnation enthalpy should conserve, as there are not interactions with rotor at inlet or exit.
r
drVdh 2
0222
2222
0
VVV
dr
drVdh rf
0222
22220
VVV
dr
d
r
V
dr
dh rf
02
0 dr
dVV
dr
dVV
dr
dVV
r
V
dr
dh rr
ff
Radial component of velocity should be constant (zero) along radial direction for radial equilibrium of flow.
02
0 dr
dVV
dr
dVV
r
V
dr
dh ff
gzUVhU
hIRothalpy bladeblade
rel 0
2
,0 2:
Constant in a turbo-machine along meridonial Plane
0
2
12
dr
rVd
r
V
dr
dV f
Stagnation enthalpy is Constant in a turbo-machine along radial direction at intake and discharge.
Twisted Blades for Large Turbines
Lessons from Nature
• In the case of a vortex, the flow field is purely tangential.
ziiW ln2
The complex potential function:
THE VORTEX
•Free Vortex Whirl:
•Forced Vortex Whirl :
General Rules for Selection of Whirl Component
r
CV
constantfV
rCV
221C rCV f
0
2
12
dr
rVd
r
V
dr
dV f
More complex Models
• Weighted mean of free and forced vortices
• General Whirl Distribution
Inlet Exit
Radial Variation of Flow Velocity in Real Machine
Intake
Discharge
Radial Variation of Whirl Velocity
Intake
Discharge
Radial Variation of Mass flow rate
Intake
Discharge